Periodic Electric Current in a Telephone Cable. 223 
equations LV 
dx eat, S68 A Ae Ee Oe (9) 
d?7I ; 
de =) an 2 (10) 
as the differential equations for the potential and current at 
any point in the cable. 
An obyious solution of the above equations is 
rales ETT) 
jak P +Pz —Pxz ¢ 
I= aE —be ie Ca ie Pe) 
where a and 0 are constants of integration, and ¢ is the base 
of Napierian logarithms. The quantity P is clearly a complex 
quantity and can be expressed in the form a+ 78. 
Hence if va + jplr/ K +jpC=a+ 78, we must have :— 
2+ P= /P4tpl VK +pe 
and a? — 9p? =RK—p°LC, 
. 2a? = /(R?+p?L?) (K?+ pC?) +(RK—p’LC), (18) 
2B?= V/(R? + p?L?)(K?+p’C?)—(RK—p*?LC. (14) 
The above equations are well-known expressions. We 
shall call 2 and 8 the secondary constants of the cable. 
Returning, then, to the case of the semi-infinite cable, we 
shall take the origin of coordinates at the end at which the 
simple periodic electromotive force of maximum value E is 
applied. 
Hence for x=0 we have V=KH, and for r=x we have 
V=0. 
Since, however, in obtaining the original differential 
equations, we reckoned 2 in the direction in which potential 
and current increase in value, we have to write —«# for # in 
the general solutions (11) and (12) before we can apply 
them to the above particular case. Making this change and 
determining the constants aand b from the terminal conditions, 
we arrive at the particular solutions of (9) and (10) applicable 
to the case of a semi-infinite cable having a periodic electro- 
motive force E applied at one end. ‘The potential V and the 
current I at any point in the cable at a distance x from 
the generator end is then given by the equations 
V=Ke ™, sie (15) 
ae VK +jpC (16) 
VR+jpL — 
